Finite Group Databasebeta

← #37554827

Subgroups — #37554827

Order 23488.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19154578C211yesmaximal, characteristic, complemented
2C22×D1468C211yesmaximal, characteristic, complemented
3#26044250C211yesmaximal, characteristic, complemented
4#26044246C212yesmaximal, complemented
5#26044236C211yesmaximal, characteristic, complemented
6#26044236C212yesmaximal, complemented
7#26044231C212yesmaximal, complemented
8#26044223C211yesmaximal, characteristic, complemented
9C4×C22⋊D367C211yesmaximal, characteristic, complemented
10C22⋊C4×D367C212yesmaximal, complemented
11C22⋊Q8×C367C211yesmaximal, characteristic, complemented
12C23×D367C2212yes—
13#19154578—24no—
14C2×D1468—24no—
15C2×D1468C2212yes—
16C2×D1468C2211yescharacteristic
17C2×D1468C2211yescharacteristic
18C2×D1468C2212yescomplemented
19C2×D1468C2212yescomplemented
20C2×C22⋊D367C2211yescharacteristic
21C2×C4×D367C2212yescomplemented
22C2×C4×D367C2211yescharacteristic, complemented
23C2×C4×D367C2212yescomplemented
24Dic367⋊C4C2211yescharacteristic, complemented
25C22⋊Dic367C2211yescharacteristic
26(C2×D367)⋊C4C2212yescomplemented
27(C2×D367)⋊C4C2212yescomplemented
28(C2×D367)⋊C4C2211yescharacteristic, complemented
29(C2×D367)⋊C4C2212yescomplemented
30(C2×D367)⋊C4C2212yescomplemented
31C4×Dic367C2211yescharacteristic, complemented
32C2×Q8×C367C2211yescharacteristic, complemented
33C22×C4×C367C2211yescharacteristic
34C4⋊C4×C367C2211yescharacteristic, complemented
35C4⋊C4×C367C2212yescomplemented
36C22⋊C4×C367C2212yes—
37C22×D367—24no—
38C22×D367—24no—
39C22×D367—42no—
40C22×D367C2312yes—
41C22×D367C2311yescharacteristic
42C22×D367C2312yes—
43D1468—24no—
44D1468—24no—
45D1468—22no—
46D1468—42no—
47D1468D414yescomplemented
48C22⋊D367—22no—
49C2×Dic367C2311yescharacteristic
50C2×Dic367C2311yescharacteristic
51C2×Dic367C2311yescharacteristic
52C4×D367—24no—
53C4×D367—22no—
54C4×D367—42no—
55C23×C367C2311yescharacteristic
56Q8×C367—22no—
57C2×C4×C367—22no—
58C2×C4×C367C2312yes—
59C2×C4×C367C2312yes—
60C2×C4×C367C2311yescharacteristic
61C2×C4×C367C2311yescharacteristic
62C22×C367—21no—
63C22×C367—21no—
64C22×C367#4112yescomplemented
65C22×C367C2411yescharacteristic
66C2×D367—22no—
67C2×D367—22no—
68C2×D367—21no—
69C2×D367—22no—
70C2×D367—21no—
71C2×D367—22no—
72C2×D367—21no—
73C2×D367—22no—
74C2×D367—48no—
75C2×D367C2×D414yes—
76C4×C367—21no—
77C4×C367—22no—
78C4×C367—22no—
79C4×C367C2×D412yes—
80Dic367—21no—
81Dic367—21no—
82Dic367—21no—
83C2×C367—22no—
84C2×C367C22×D411yescharacteristic
85C2×C367C2×#4111yescharacteristic
86C2×C367H(5,𝔽2)11yescharacteristic
87D367—24no—
88D367—41no—
89D367—42no—
90C367#54611yescharacteristic, complemented, Sylow
91#546—3671nomaximal, Sylow
92C2×#41—3671no—
93C22×D4—3671no—
94#124—3671no—
95#123—3672no—
96C22⋊Q8D36711yescharacteristic, complemented
97#121—3671no—
98#121—3672no—
99#120—3672no—
100C4×D4—3671no—
101C4×D4—3671no—
102C2×C22⋊C4—3672no—
103C24—3672no—
104#41—7344no—
105C2×Q8C2×D36711yescharacteristic, complemented
106C2×D4—3671no—
107C2×D4—3671no—
108C2×D4—3672no—
109C2×D4—3672no—
110C2×D4—3672no—
111C2×D4—3671no—
112C2×D4—7344no—
113C22×C4—3672no—
114C22×C4—3672no—
115C22×C4—3671no—
116C22×C4C2×D36711yescharacteristic
117C4⋊C4—3671no—
118C4⋊C4C2×D36711yescharacteristic, complemented
119C4⋊C4C2×D36712yescomplemented
120C22⋊C4—3671no—
121C22⋊C4—3672no—
122C22⋊C4—3672no—
123C22⋊C4—3672no—
124C22⋊C4—3671no—
125C22⋊C4—3672no—
126C22⋊C4C2×D36712yes—
127C42—3671no—
128C23—3672no—
129C23—3672no—
130C23—3671no—
131C23—14682no—
132C23—7344no—
133C23—7344no—
134C23C22×D36711yescharacteristic
135Q8—22no—
136D4—3674no—
137D4—14682no—
138D4—7344no—
139D4—7344no—
140D4—7342no—
141D4—7342no—
142C2×C4—22no—
143C2×C4—3671no—
144C2×C4—3671no—
145C2×C4—3671no—
146C2×C4—14682no—
147C2×C4—7344no—
148C2×C4—7342no—
149C2×C4C22×D36712yes—
150C2×C4C22×D36712yes—
151C2×C4C22×D36711yescharacteristic
152C2×C4C22×D36711yescharacteristic
153C22—21no—
154C22—21no—
155C22—3674no—
156C22—14688no—
157C22—7341no—
158C22—7342no—
159C22—7341no—
160C22—7342no—
161C22—7341no—
162C22—7342no—
163C22—7342no—
164C22—7342no—
165C22#1915457812yescomplemented
166C22C23×D36711yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—7341no—
171C4—7341no—
172C4—7341no—
173C4D4×D36712yes—
174C2—22no—
175C2—14682no—
176C2—14681no—
177C2—7344no—
178C2C2×D4×D36711yescharacteristic
179C2C2×#1915457811yescharacteristic
180C2#2604430711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D367
commutator, socle, cosocleC22×C367C24
FittingC22⋊Q8×C367C2
hypercenterC22⋊Q8D367
hypocenterC367#546
LCSG ▷ C22×C367 ▷ C367
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19154578C2semidirect
2. C22×D1468C2semidirect
3. #26044250C2semidirect
4. #26044246C2semidirect
5. #26044236C2semidirect
6. #26044236C2semidirect
7. #26044231C2semidirect
8. #26044223C2semidirect
9. C4×C22⋊D367C2semidirect
10. C22⋊C4×D367C2semidirect
11. C22⋊Q8×C367C2semidirect
18. C2×D1468C22semidirect
19. C2×D1468C22semidirect
21. C2×C4×D367C22semidirect
22. C2×C4×D367C22semidirect
23. C2×C4×D367C22semidirect
24. Dic367⋊C4C22semidirect
26. (C2×D367)⋊C4C22semidirect
27. (C2×D367)⋊C4C22semidirect
28. (C2×D367)⋊C4C22semidirect
29. (C2×D367)⋊C4C22semidirect
30. (C2×D367)⋊C4C22semidirect
31. C4×Dic367C22semidirect
32. C2×Q8×C367C22semidirect
34. C4⋊C4×C367C22semidirect
35. C4⋊C4×C367C22semidirect
47. D1468D4semidirect
64. C22×C367#41semidirect
90. C367#546semidirect
96. C22⋊Q8D367semidirect
105. C2×Q8C2×D367semidirect
118. C4⋊C4C2×D367semidirect
119. C4⋊C4C2×D367semidirect
165. C22#19154578semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#191545782
C22×D14682
#260442502
#260442462
#260442362
#260442312
#260442232
C4×C22⋊D3672
C22⋊C4×D3672
C22⋊Q8×C3672
#546367